| L(s) = 1 | − 2·5-s + 2·11-s − 2·23-s + 3·25-s + 2·31-s + 2·37-s − 2·41-s + 2·43-s + 2·47-s − 4·55-s + 2·67-s + 2·71-s + 2·83-s − 2·97-s + 2·101-s + 2·103-s + 2·107-s + 4·115-s + 121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 4·155-s + ⋯ |
| L(s) = 1 | − 2·5-s + 2·11-s − 2·23-s + 3·25-s + 2·31-s + 2·37-s − 2·41-s + 2·43-s + 2·47-s − 4·55-s + 2·67-s + 2·71-s + 2·83-s − 2·97-s + 2·101-s + 2·103-s + 2·107-s + 4·115-s + 121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 4·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.171809959\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.171809959\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + T^{4} \) |
| 17 | $C_2^2$ | \( 1 + T^{4} \) |
| 19 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + T^{4} \) |
| 59 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + T^{4} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.777362199449354981429506453265, −8.676756732960514877934253253136, −8.105924723056489964935689644099, −8.083620340777924822969344459628, −7.49471436028805230598606804052, −7.32499834147698904851577333336, −6.66332884956098550923252036595, −6.58514961415288816056875635274, −6.01944252265497369817833712457, −5.87215468540369274275750694911, −4.85674664331473096631339434902, −4.78124692557483828853010400450, −4.11135242795724782348335774594, −4.09223761042493501301846388077, −3.53514929606978651006794279744, −3.40691136244679509257435966282, −2.36223415565357696474986307221, −2.29999990034130846099545278391, −1.02092110057465979836534528953, −0.884774419936471199632935260158,
0.884774419936471199632935260158, 1.02092110057465979836534528953, 2.29999990034130846099545278391, 2.36223415565357696474986307221, 3.40691136244679509257435966282, 3.53514929606978651006794279744, 4.09223761042493501301846388077, 4.11135242795724782348335774594, 4.78124692557483828853010400450, 4.85674664331473096631339434902, 5.87215468540369274275750694911, 6.01944252265497369817833712457, 6.58514961415288816056875635274, 6.66332884956098550923252036595, 7.32499834147698904851577333336, 7.49471436028805230598606804052, 8.083620340777924822969344459628, 8.105924723056489964935689644099, 8.676756732960514877934253253136, 8.777362199449354981429506453265
